argon - 39 has a half - life of 269 years. how long will it take for 52.5 g of a 60.0 g sample to decay to…

Explanation:

Step1: Calculate remaining amount

The initial amount $N_0 = 60.0$ g and the amount that has decayed is 52.5 g. So the remaining amount $N=60.0 - 52.5=7.5$ g.

Step2: Use radioactive - decay formula

The radioactive - decay formula is $N = N_0(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, where $T_{1/2}$ is the half - life, $t$ is the time elapsed, $N$ is the remaining amount, and $N_0$ is the initial amount. Substitute $N = 7.5$ g, $N_0 = 60.0$ g, and $T_{1/2}=269$ years into the formula: $7.5=60.0(\frac{1}{2})^{\frac{t}{269}}$. First, divide both sides by 60.0: $\frac{7.5}{60.0}=(\frac{1}{2})^{\frac{t}{269}}$. $\frac{7.5}{60.0}=\frac{1}{8}$, so $\frac{1}{8}=(\frac{1}{2})^{\frac{t}{269}}$. Since $\frac{1}{8}=(\frac{1}{2})^3$, we have $(\frac{1}{2})^3=(\frac{1}{2})^{\frac{t}{269}}$. Then, set the exponents equal to each other: $3=\frac{t}{269}$.

Step3: Solve for $t$

Multiply both sides by 269: $t = 3\times269 = 807$ years.

Answer:

807